Numerical Methods for Nonlinear Partial Differential Equations, with applications to Optimal Transportation, and Geometric Data Reduction

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A fundamental change is taking place in the role of applied and computational mathematics.****Across science, technology, commerce, and medicine our ability to collect and store data is increasing dramatically. These developments call for a change in our understanding of data and demand the development of new tools. The techniques of computational mathematics are uniquely suited for building these tools. ****I propose to build new algorithms for large data sets and apply these algorithms to solve important real-world problems. The primary objectives are to: ****1. Develop an extension of my algorithm for solving the Optimal Transportation (OT) problem. One of the fundamental tools of mathematics is impose a metric structure on complex sets. The Wasserstein distance does so for the set of probability measures. However, evaluating this distance is not given explicitly: it is obtained as the solution of an infinite dimensional variational problem, the Optimal Transportation problem. Building good methods for solving this problem allow this distance to be computed.***2. Develop an algorithm for data reduction using extremal points. Given a scatterplot of two dimensional points, the extremal values are the farthest out from the centre. Mathematically, the convex hull is used to represent the extremal values. There are already algorithms for finding convex hulls, but they break down when the number of points, or the dimension of the points, is too large. The proposed research will develop algorithms to find extremal points of large set of high dimensional data.***3. Develop an algorithm, Multidimensional Quicksort, for anomaly detection. You may perform anomaly detection when you check your monthly credit card bill. For example, looking for items that are unusual, either because of the amount charged, or because the vendor is not recognized. The proposed research will develop algorithms to perform anomaly detection on huge amounts of data very quickly. The data is simplified by using a density function, which represents the pattern of the types of anomalies we are looking for. The process of sorting the points is replaced by solving an equation involving the density function. Once we have the solution of the equation, we can use it to quickly find the anomalies.***4. Continue foundational work building of nonlinear Partial Differential Equation (PDE) solvers. ****The corresponding expected outcomes include: ***1.(a) An effective method to compare documents or images.*** (b) A geometry-sensitive method to perform registration of shapes or images.*** (c) A state-of-the-art image warping tool, to be used by several top surgeons at US hospitals. ***2.(a) A data reduction tool for finding extreme values of high dimensional data.*** (b) Improved off-line and potential real-time performance monitoring for helicopters. ****3. Streaming detection of stock anomalies. ****4. Effective solvers for new equations, leading to future outcomes like 1-3 above.**